Careful measurements have been made of the critical currents of a number of superconducting weak links formed using niobium point contacts. Values for these critical currents as functions of temperature have been derived both from magnetization curves obtained when the links were included in a structure with a two-hole SQUID geometry and from current-voltage measurements. The data were corrected for the effects of thermal activation. Discrepancies between the results and theoretical predictions were attributed to non-sinusoidal current-phase relationships; magnetization measurements were performed on very weak links which produced direct evidence for this effect. The standard theoretical model was modified to include a second-harmonic term in the current-phase relationship and the resulting predictions agreed well with the experimental results. No evidence was obtained for macroscopic quantum tunnelling in our devices, which is consistent with the prediction that this effect should be suppressed by the electromagnetic properties of the superconducting structure surrounding the point contact.